Pricing financial instruments is a complex art requiring substantial expertise and experience. Trading financial instruments, such as options, involves a sophisticated process of pricing typically performed by a trader.
The term “option” in the context of the present application is broadly defined as any financial instrument having option-like properties, e.g., any financial derivative including an option or an option-like component. This category of financial instruments may include any type of option or option-like financial instrument, relating to some underlying asset. Assets as used in this application include anything of value; tangible or non-tangible, financial or non-financial, for example, stocks; currencies; commodities, e.g., oil, metals, or sugar; interest rates; forward-rate agreements (FRA); swaps; futures; bonds; weather, e.g., the temperature at a certain area; electricity; gas emission; credit; mortgages; indices; and the like. For example, as used herein, options range from a simple Vanilla option on a single stock and up to complex convertible bonds whose convertibility depends on some key, e.g., the weather.
The term “Exchange” in the context of the present application relates to any one or more exchanges throughout the world, and includes all assets/securities, which may be traded in these exchanges. The terms “submit a price to the exchange”, “submit a quote to the exchange”, and the like generally refer to actions that a trader may perform to submit a bid and/or offer prices for trading in the exchange. The price may be transferred from the trader to the exchange, for example, by a broker, by online trading, on a special communication network, through a clearing house system, and/or using in any other desired system and/or method.
The price of an asset for immediate, e.g., 1 or 2 business days, delivery is called the spot price. For an asset sold in an option contract, the strike price is the agreed upon price at which the deal is executed if the option is exercised. For example, a stock option involves buying or selling a stock. The spot price is the current stock price on the exchange in which is the stock is traded. The strike price is the agreed upon price to buy/sell the stock if the option is exercised.
To facilitate trading of options and other financial instruments, a market maker suggests a bid price and offer price (also called ask price) for a certain option. The bid price is the price at which the market maker is willing to purchase the option and the offer price is the price at which the market maker is willing to sell the option. As a market practice, a first trader interested in a certain option may ask a second trader for a quote, e.g., without indicating whether the first trader is interested to buy or to sell the option. The second trader quotes both the bid and offer prices, not knowing whether the first trader is interested in selling or buying the option. The market maker may earn a margin by buying options at a first price and selling them at a second price, e.g., higher than the first price. The difference between the offer and bid prices is referred to as bid-offer spread.
A call option is the right to buy an asset at a certain price (“the strike”) at a certain time, e.g., on a certain date. A put option is the right to sell an asset at a strike price at a certain time, e.g., on a certain date. Every option has an expiration time in which the option ceases to exist. Prior to the option expiration time, the holder of the option may determine whether or not to exercise the option, depending on the prevailing spot price for the underlying asset. If the spot price at expiration is lower than the strike price, the holder will choose not to exercise the call option and lose only the cost of the option itself. However, if the strike is lower than the spot, the holder of the call option will exercise the right to buy the underlying asset at the strike price making a profit equal to the difference between the spot and the strike prices. The cost of the option is also referred to as the premium.
A forward rate is defined as the predetermined rate or price of an asset, at which an agreed upon future transaction will take place. The forward rate may be calculated based on a current rate of the asset, a current interest rate prevailing in the market, expected dividends (for stocks), cost of carry (for commodities), and/or other parameters depending on the underlying asset of the option.
An at-the-money forward option (ATM) is an option whose strike is equal to the forward rate of the asset. In some fields, the at-the-money forward options are generically referred to as at-the-money options, as is the common terminology in the commodities and interest rates options. The at the money equity options are actually the at the money spot, i.e. where the strike is the current spot rate or price.
An in-the-money call option is a call option whose strike is below the forward rate of the underlying asset, and an in the-money put option is a put option whose strike is above the forward rate of the underlying asset. An out-of-the-money call option is a call option whose strike is above the forward rate of the underlying asset, and an out-of-the-money put option is a put option whose strike is below the forward rate of the underlying asset.
An exotic option, in the context of this application, is a generic name referring to any type of option other than a standard Vanilla option. While certain types of exotic options have been extensively and frequently traded over the years, and are still traded today, other types of exotic options had been used in the past but are no longer in use today. Currently, the most common exotic options include “barrier” options, “digital” options, “binary” options, “partial barrier” options (also known as “window” options), “average” options, “compound” options and “quanto” options. Some exotic options can be described as a complex version of the standard (Vanilla) option. For example, barrier options are exotic options where the payoff depends on whether the underlying asset's price reaches a certain level, hereinafter referred to as “trigger”, during a certain period of time. The “pay off” of an option is defined as the cash realized by the holder of the option upon its expiration. There are generally two types of barrier options, namely, a knock-out option and a knock-in option. A knock-out option is an option that terminates if and when the spot reaches the trigger. A knock-in option comes into existence only when the underlying asset's price reaches the trigger. It is noted that the combined effect of a knock-out option with strike K and trigger B and a knock-in option with strike K and trigger B, both having the same expiration, is equivalent to a corresponding Vanilla option with strike K. Thus, knock-in options can be priced by pricing corresponding knock-out and vanilla options. Similarly, a one-touch option can be decomposed into two knock-in call options and two knock-in put options, a double no-touch option can be decomposed into two double knock-out options, and so on. It is appreciated that there are many other types of exotic options known in the art.
Certain types of options, e.g., Vanilla options, are commonly categorized as either European or American. A European option can be exercised only upon its expiration. An American option can be exercised at any time after purchase and before expiration. For example, an American Vanilla option has all the properties of the Vanilla option type described above, with the additional property that the owner can exercise the option at any time up to and including the option's expiration date. As is known in the art, the right to exercise an American option prior to expiration makes American options more expensive than corresponding European options.
Generally in this application, the term “Vanilla” refers to a European style Vanilla option. European Vanilla options are the most commonly traded options; they are typically traded over the counter (OTC). American Vanilla options are more popular in the exchanges and, in general, are more difficult to price.
U.S. Pat. No. 5,557,517 (“the '517 patent”) describes a method of pricing American Vanilla options for trading in a certain exchange. This patent describes a method of pricing Call and Put American Vanilla options, where the price of the option depends on a constant margin or commission required by the market maker.
The method of the '517 patent ignores data that may affect the price of the option, except for the current price of the underlying asset and, thus, this method can lead to serious errors, for example, an absurd result of a negative option price. Clearly, this method does not emulate the way American style Vanilla options are priced in real markets.
The Black-Scholes (BS) model (developed in 1973) is a widely accepted method for valuing options. This model calculates a theoretical value (TV) for options based on the probability of the payout, which is commonly used as a starting point for approximating option prices. This model is based on a presumption that the change in the spot price of the asset generally follows a Brownian motion, as is known in the art. Using such Brownian motion model, known also as a stochastic process, one may calculate the theoretical price of any type of financial derivative, either analytically or numerically. For example, it is common to calculate the theoretical price of complicated financial derivatives through simulation techniques, such as the Monte-Carlo method, introduced by Boyle in 1977. Such techniques may be useful in calculating the theoretical value of an option, provided that a computer being used is sufficiently powerful to handle all the calculations involved. In the simulation method, the computer generates many propagation paths for the underlying asset, starting at the trade time and ending at the time of the option expiry. Each path is discrete and generally follows the Brownian motion probability, but may be generated as densely as necessary by reducing the time lapse between each move of the underlying asset. Thus, if the option is path-dependant, each path is followed and only the paths that satisfy the conditions of the option are taken into account. The end results of each such path are summarized and lead to the theoretical price of the derivative.
The original Black-Scholes model was derived for calculating theoretical prices of European Vanilla options, where the price of the option is described by a relatively simple formula. However, it should be understood that any reference in this application to the Black-Scholes model refers to use of the Black-Scholes model or any other suitable model for evaluating the behavior of the underlying asset, e.g., assuming a stochastic process (Brownian motion), and/or for evaluating the price of any type of option, including exotic options. Furthermore, this application is general and independent of the way in which the theoretical value of the option is obtained. It can be derived analytically, numerically, using any kind of simulation method or any other technique available.
For example, U.S. Pat. No. 6,061,662 (“the '662 patent”) describes a method of evaluating the theoretical price of an option using a Monte-Carlo method based on historical data. The simulation method of the '662 patent uses stochastic historical data with a predetermined distribution function in order to evaluate the theoretical price of options. Examples is the '662 patent are used to illustrate that this method generates results which are very similar to those obtained by applying the Black-Scholes model to Vanilla options. Unfortunately, methods based on historical data alone are not relevant for simulating financial markets, even for the purpose of theoretical valuation. For example, one of the most important parameters used for valuation of options is the volatility of the underlying asset, which is a measure for how the price and/or rate of the underlying asset may fluctuate. It is well known that the financial markets use a predicted, or an expected, value for the volatility of the underlying assets, which often deviates dramatically from the historical data. In market terms, expected volatility is often referred to as “implied volatility”, and is differentiated from “historical volatility”. For example, the implied volatility tends to be much higher than the historical volatility of the underlying asset before a major event, such as risk of war, and in anticipation of or during a financial crisis.
It is appreciated by persons skilled in the art that the Black-Scholes model is a limited approximation that may yield results very far from real market prices and, thus, corrections to the Black-Scholes model must generally be added by traders. For example, in the Foreign Exchange (FX) Vanilla market, and in commodities, the market trades in volatility terms and the translation to option price is performed through use of the Black-Scholes formula. In fact, traders commonly refer to using the Black-Scholes model as “using the wrong volatility with the wrong model to get the right price”.
In order to adjust the BS price, in the Vanilla market, traders use different volatilities for different strikes, i.e., instead of using one volatility per asset per expiration date, as is required by the BS model, a trader may use different volatility values for a given asset depending on the strike price. This adjustment is known as volatility “smile” adjustment. The origin of the term “smile”, in this context, is the typical shape of the volatility vs. strike, which is similar to a flat “U” shape (smile).
The phrase “market price of an option” is used herein to distinguish between the single value produced by some known models, such as the Black-Scholes model, and the actual bid and offer prices traded in the real market. For example, for some options, the market bid side may be twice the Black-Scholes model price and the offer side may be three times the Black-Scholes model price.
Many exotic options are characterized by discontinuity of the payout and, therefore, a discontinuity in some of the risk parameters near the trigger(s). This discontinuity prevents an oversimplified model such as the Black-Scholes model from taking into account the difficulty in risk-managing the option. Furthermore, due to the peculiar profile of some exotic options, there may be significant transaction costs associated with re-hedging some of the risk factors. Existing models, such as the Black-Scholes model, completely ignore such risk factors.
Several options pricing models were introduced since 1973, but none of these models was able to replicate the market prices universally and/or consistently. The most famous pricing models include, the stochastic volatility model, which assumes that the volatility itself is another stochastic process correlated with the underlying process; the local volatility model, where the volatility is a function of time and the underlying asset; and Libor based models, such as BGM, which generate the swaption prices from the Libor rates which are correlated stochastic processes.
Many factors may be taken into account in calculating option prices and corrections. The term “Factor” is used herein broadly as any quantifiable or computable value relating to the subject option. Some of the notable factors are defined as follows.
Volatility (“Vol”) is a measure of the fluctuation of the return realized on an asset, e.g., a daily return. An indication of the order of magnitude the volatility can be obtained by historical volatility, i.e., the standard deviation of the daily return of the assets for a certain past period.
However, the markets trade based on a volatility that reflects the market expectations of the standard deviation in the future. The volatility reflecting market expectations is called implied volatility. In order to buy/sell volatility one commonly trades Vanilla options. For example, in the foreign exchange market, the implied volatilities of ATM Vanilla options for frequently used option dates and currency pairs are available to users in real-time, e.g., via screens such as REUTERS, Bloomberg or directly from FX option brokers.
Volatility smile, as discussed above, relates to the behavior of the implied volatility with respect to the strike, i.e., the implied volatility as a function of the strike, where the implied volatility for the ATM strike is the given ATM volatility in the market. Typically the plot of the implied volatility as a function of the strike shows a minimum that looks like a smile. For example, usually in equity options the minimum volatility is below the ATM strike.
Delta is the rate of change in the price of an option in response to changes in the price of the underlying asset; in other words, it is a partial derivative of the option price with respect to the spot. For example, a 25 delta call option is defined as follows: if against buying the option on one unit of the underlying asset, 0.25 units of the underlying asset are sold, then for small changes in the underlying asset price, assuming all other factors are unchanged, the total change in the price of the option and the profit or loss generated by holding 0.25 units of the asset are null.
Vega is the rate of change in the price of an option or other derivative in response to changes in volatility, i.e., the partial derivative of the option price with respect to the volatility.
Volatility Convexity is the second partial derivative of the price with respect to the volatility, i.e. the derivative of the Vega with respect to the volatility, denoted dVega/dVol.
Straddle is a strategy, which includes buying Vanilla call and put options having the same strike price and the same expiration.
At-the-money Delta neutral straddle is a straddle wherein the Delta of the call option and the Delta of the put option have the same value with opposite sign. The buyer of the at-the-money Delta neutral straddle strategy is automatically Delta-hedged (protected from small changes in the price of the underlying asset).
Risk Reversal (RR) is a strategy, which includes buying a Vanilla call option and selling a Vanilla put option with the same expiration sand the same Delta with opposite sign. In some markets, the RR corresponds to the difference between the implied volatility of a call option and a put option with the same delta (in opposite directions). Traders in the currency and/or commodity option markets generally use 25delta RR, which is the difference between the implied volatility of a 25delta call option and a 25delta put option. Thus, 25delta RR may be calculated as follows:25delta RR=implied Vol(25delta call)−implied Vol(25delta put)
The 25delta RR may correspond to a combination of buying a 25 delta call option and selling a 25 delta put option. Accordingly, the 25delta RR may be characterized by a slope of Vega of such combination with respect to spot. Thus, the price of the 25delta RR may characterize the price of the Vega slope, since practically the convexity of 25delta RR at the current spot is close to zero. Therefore, the 25delta RR as defined above may be used to price the slope dVega/dspot.
Strangle is a strategy of buying call and put options with the same expiration. In some applications, the call and put options may have the same Delta with opposite signs. The strangle price can be presented as the average of the implied volatility of the call and put options. For example:25delta strangle=0.5(implied Vol(25delta call)+implied Vol(25delta put))
The 25delta strangle may be characterized by practically no slope of Vega with respect to spot at the current spot, but a lot of convexity, i.e., a change of Vega when the volatility changes. Therefore, it is used to price convexity.
Since the at-the-money Vol may be known, it is more common to quote the butterfly strategy, in which one buys one unit of the strangle and sells 2 units of the ATM 25 option. In some assets, the strangle/butterfly is quoted in terms of volatility. For example:25delta butterfly=0.5*(implied Vol(25delta call)++implied Vol(25delta put))−ATM Vol
The reason it is more common to quote the butterfly rather than the strangles is that butterfly provides a strategy with almost no Vega but significant convexity. Since butterfly and strangle are related through the ATM volatility, which may be known, they may be used interchangeably. The 25delta put and the 25delta call can be determined based on the 25delta RR and the 25delta strangle. The ATM volatility, 25 delta risk reversal and/or the 25 delta butterfly may be referred to, for example, as the “Volatility Parameters”. The Volatility Parameters may include any additional and/or alternative parameters and/or factors.
Bid/offer spread is the difference between the bid price and the offer price of a financial derivative. In the case of options, the bid/offer spread may be expressed, for example, either in terms of volatility or in terms of the price of the option. For example, the bid/ask spread of exchange traded options is quoted in price terms (e.g., cents, etc). The bid/offer spread of a given option depends on the specific parameters of the option. In general, the more difficult it is to manage the risk of an option, the wider is the bid/offer spread for that option.
In order to quote a price, traders typically try to calculate the price at which they would like to buy an option (i.e., the bid side) and the price at which they would like to sell the option (i.e., the offer side). Many traders have no computational methods for calculating the bid and offer prices, and so traders typically rely on intuition, experiments involving changing the factors of an option to see how they affect the market price, and past experience, which is considered to be the most important tool of traders.
One dilemma commonly faced by traders is how wide the bid/offer spread should be. Providing too wide a spread reduces the ability to compete in the options market and is considered unprofessional, yet too narrow a spread may result in losses to the trader. In determining what prices to provide, traders need to ensure that the bid/offer spread is appropriate. This is part of the pricing process, i.e., after the trader decides where to place the bid and offer prices, he/she needs to consider whether the resultant spread is appropriate. If the spread is not appropriate, the trader needs to change either or both of the bid and offer prices in order to show the appropriate spread.
Option prices that are quoted in exchanges typically have a relatively wide spread compared to their bid/ask spread in the OTC market, where traders of banks typically trade with each other through brokers. In addition the exchange price typically corresponds to small notional amounts of options (lots). A trader may sometimes change the exchange price of an option by suggesting a bid price or an offer price with a relatively small amount of options. This may result in the exchange prices being distorted in a biased way.
In contrast to the exchanges, the OTC option market has a greater “depth” in terms of liquidity. Furthermore, the options traded in the OTC market are not restricted to the specific strikes and expiration dates of the options traded in the exchanges. In addition, there are many market makers, which quote bid/offer prices, which are totally different from the bid/offer prices in the exchange.
One of the reasons that exchange prices of options are quoted with a wide spread is that the prices of options corresponding to many different strikes, and many different dates may change very frequently, e.g., in response to each change in the price of the underlying assets. As a result, the people that provide the bid and ask prices to the exchange have to constantly update a large number of bid and ask prices simultaneously, e.g., each time the price of the underlying assets changes. In order to avoid this tedious activity, it is mostly preferred to use “safe” bid and ask prices, which will not need to be frequently updated.